Consider the complement \(\overline{G}\)  of \(G\) . Since \(\overline{G}\)  is a chordal graph, it has a simplicial vertex [1]}, say \(x\in V\) . This means that the set \(N_{\overline{G}}(x)\)  induces a complete subgraph of \(\overline{G}\) . Now, if \(u,v\in V\)  are two vertices with \(uv\notin E(\overline{G})\) , then \(V(\overline{G}-N_{\overline{G}}(x))\cap \lbrace u,v\rbrace \ne \emptyset \) . However, this latter fact is equivalent to saying that \(N_G[x]\cap \lbrace u,v\rbrace \ne \emptyset \)  whenever \(uv\in E(G)\) , i.e., the set \(N_G[x]\)  is a vertex cover for \(G\) .
